Towards a resolution of the Buchanan-Lillo conjecture

TL;DR

This paper investigates the asymptotic behavior of oscillatory solutions in a first-order delay differential equation with mixed feedback, providing new insights into the Buchanan-Lillo conjecture and characterizing the threshold between bounded and unbounded solutions.

Contribution

It characterizes the convergence of oscillatory solutions to a special periodic solution in the mixed feedback case, extending and refining the Buchanan-Lillo conjecture.

Findings

Oscillatory solutions on the threshold are asymptotic to special periodic solutions.

The results include the periodic solution of the positive feedback case.

The paper clarifies the boundary between bounded and unbounded oscillations.

Abstract

Buchanan and Lillo both conjectured that oscillatory solutions of the first-order delay differential equation with positive feedback $x^{\prime }(t)=p(t)x(\tau (t))$, $t\geq 0$, where $0\leq p(t)\leq 1$, $0\leq t-\tau (t)\leq 2.75+\ln2,t\in \mathbb{R},$ are asymptotic to a shifted multiple of a unique periodic solution. This special solution was known to be uniform for all nonautonomous equations, and intriguingly, can also be described from the more general perspective of the mixed feedback case (sign-changing $p$). The analog of this conjecture for negative feedback, $p(t)\leq0$, was resolved by Lillo, and the mixed feedback analog was recently set as an open problem. In this paper, we investigate the convergence properties of the special periodic solutions in the mixed feedback case, characterizing the threshold between bounded and unbounded oscillatory solutions, with standing assumptions that $p$ and $\tau$ are measurable, $\tau (t)\leq t$ and $\lim_{t\rightarrow \infty }\tau (t)=\infty$. We prove that nontrivial oscillatory solutions on this threshold are asymptotic (differing by $o(1)$) to the special periodic solutions for mixed feedback, which include the periodic solution of the positive feedback case. The conclusions drawn from these results elucidate and refine the conjecture of Buchanan and Lillo that oscillatory solutions in the positive feedback case $p(t)\geq0$, would differ from a multiple, translation, of the special periodic solution, by $o(1)$.